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Difference between revisions of "Unitary space"

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A [[Vector space|vector space]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955801.png" /> of complex numbers, on which there is given an [[Inner product|inner product]] of vectors (where the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955802.png" /> of two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955804.png" /> is, in general, a complex number) that satisfies the following axioms:
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A [[Vector space|vector space]] over the field $\mathbf C$ of complex numbers, on which there is given an [[Inner product|inner product]] of vectors (where the product $(a,b)$ of two vectors $a$ and $b$ is, in general, a complex number) that satisfies the following axioms:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955805.png" />;
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1) $(a,b)=\overline{(b,a)}$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955806.png" />;
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2) $(\alpha a,b)=\alpha(a,b)$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955807.png" />;
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3) $(a+b,c)=(a,c)+(b,c)$;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955808.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095580/u0955809.png" />, i.e. the scalar square of a non-zero vector is a positive real number.
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4) if $a\neq0$, then $(a,a)>0$, i.e. the scalar square of a non-zero vector is a positive real number.
  
 
A unitary space need not be finite-dimensional. In a unitary space one can, just as in Euclidean spaces, introduce the concept of orthogonality and of an orthonormal system of vectors, and in the finite-dimensional case one can prove the existence of an orthonormal basis.
 
A unitary space need not be finite-dimensional. In a unitary space one can, just as in Euclidean spaces, introduce the concept of orthogonality and of an orthonormal system of vectors, and in the finite-dimensional case one can prove the existence of an orthonormal basis.

Revision as of 17:34, 11 April 2014

A vector space over the field $\mathbf C$ of complex numbers, on which there is given an inner product of vectors (where the product $(a,b)$ of two vectors $a$ and $b$ is, in general, a complex number) that satisfies the following axioms:

1) $(a,b)=\overline{(b,a)}$;

2) $(\alpha a,b)=\alpha(a,b)$;

3) $(a+b,c)=(a,c)+(b,c)$;

4) if $a\neq0$, then $(a,a)>0$, i.e. the scalar square of a non-zero vector is a positive real number.

A unitary space need not be finite-dimensional. In a unitary space one can, just as in Euclidean spaces, introduce the concept of orthogonality and of an orthonormal system of vectors, and in the finite-dimensional case one can prove the existence of an orthonormal basis.


Comments

References

[a1] W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 338
[a2] W.H. Greub, "Linear algebra" , Springer (1975) pp. Chapt. XI
How to Cite This Entry:
Unitary space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_space&oldid=31532
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article